$p$-adic field 2.2.8.62a1.1620

• Label: 2.2.8.62a1.1620
• Base: \(\Q_{2}\)
• Degree: \(16\)
• e: \(8\)
• f: \(2\)
• c: \(62\)
• Galois group: $Q_{16}:C_2$ (as 16T50)

Defining polynomial

$( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + 16 ( x^{2} + x + 1 ) + 32 x + 2$

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$16$
Ramification index $e$:	$8$
Residue field degree $f$:	$2$
Discriminant exponent $c$:	$62$
Discriminant root field:	$\Q_{2}$
Root number:	$1$
$\Aut(K/\Q_{2})$:	$Q_8$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:	$[3, 4, 5]$
Visible Swan slopes:	$[2,3,4]$
Means:	$\langle1, 2, 3\rangle$
Rams:	$(2, 4, 8)$
Jump set:	$[1, 3, 7, 15]$
Roots of unity:	$6 = (2^{ 2 } - 1) \cdot 2$

Intermediate fields

$\Q_{2}(\sqrt{5})$, $\Q_{2}(\sqrt{-2})$, $\Q_{2}(\sqrt{-2\cdot 5})$, 2.2.2.6a1.1, 2.1.4.11a1.4 x2, 2.1.4.11a1.7 x2, 2.2.4.22a1.13

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Canonical tower

Unramified subfield:	$\Q_{2}(\sqrt{5})$ $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{2} + x + 1 \)
Relative Eisenstein polynomial:	\( x^{8} + 8 x^{6} + 16 x^{5} + 16 x^{3} + 8 x^{2} + 16 x + 32 t + 2 \) $\ \in\Q_{2}(t)[x]$

Ramification polygon

Residual polynomials:	$z^4 + 1$,$z^2 + 1$,$z + 1$
Associated inertia:	$1$,$1$,$1$
Indices of inseparability:	$[24, 16, 8, 0]$

Invariants of the Galois closure

Galois degree:	$32$
Galois group:	$Q_{16}:C_2$ (as 16T50)
Inertia group:	Intransitive group isomorphic to $\OD_{16}$
Wild inertia group:	$\OD_{16}$
Galois unramified degree:	$2$
Galois tame degree:	$1$
Galois Artin slopes:	$[2, 3, 4, 5]$
Galois Swan slopes:	$[1,2,3,4]$
Galois mean slope:	$4.0$
Galois splitting model:	$x^{16} - 24 x^{14} + 252 x^{12} - 1440 x^{10} + 4770 x^{8} - 8640 x^{6} + 5832 x^{4} + 2592 x^{2} + 324$